Message capacity and claim wording set the transition points of collective truth-finding in language-model networks

Makoto Fukushima

cs.MASep 15, 2026

Message capacity and claim wording set the transition points of collective truth-finding in language-model networks

Authors: Makoto Fukushima

Organizations: Research Division, Honda Research Institute Japan Co., Ltd., Wako, Saitama, Japan

Abstract

Whether human or large language model (LLM), an agent in a discussion reads only a few of the others' contributions, bounded by cognition, context, or cost. LLM collectives can settle on a wrong consensus even when a majority starts out correct; we ask how far that reading bound alone decides the outcome. We model the bound with one number, the message capacity, which sets how many of the others' messages an agent reads, and generate the communication network from it. Over 31,824 randomized queries, we found that an 8-billion-parameter model's judgment of a claim effectively reduces to a logistic function of a weighted sum of its inbox, the update rule of a stochastic binary neuron with divisively normalized weights. From these weights and the network's degree statistics alone, the wrong consensus should become unreachable from any start once agents read, on average, fewer than 6.4 of their 31 sources. In 1,414 episodes with assigned starts the prediction failed: the correct side won in fewer than 50% of episodes from every start, and in only 28-45% when 75% of agents started correct. The failure traces to the field, the threshold that a claim's wording sets for the agent's answer before any message is read: the experimental claims' fields lay below the calibration mean, and with each claim's own field the same weights reproduce the outcomes. Reversing the wording showed that the threshold follows what a claim asserts, not whether it is true. On a second 8B model the pipeline predicts claim-dependent bistability; transition points appeared where computed, and an eight-claim calibration matched in 15 of 16 conditions. At 70B the assertion bias is not detected. Thus a collective's fate is largely set by two single-agent measurements: the threshold a claim's wording sets, and the message capacity that sets the transition point.

Figures & tables

Figure 1: Crowding-generated networks and the prediction pipeline. (a) In-degree distribution

P_{\alpha,N}(k)

generated by the acceptance process

e^{-\alpha r}

(inset) at three crowding levels,

N=32

. (b) Every quantity used to predict the collective outcome is measured on single agents or computed from the reduced map; nothing is fitted to collective data.

Figure 2: Critical crowding level from the reduced map. (a) Fixed points of the reduced map

G

as functions of

\alpha

for the four measured coefficient sets of El et al. (2026a) ; the interior unstable branch (open symbols) vanishes at

\alpha^{*}

. (b) Validation of the two closure levels against the fold of the full map

F

over 11 coefficient sets (filled symbols: coefficient sets measured in this study).

Figure 3: Observed correct-side rate against fitted net input under divisive and constant message weights. (a) The measured update rule: $l$ correct-side and $k-l$ wrong-side messages enter with weights $\beta_{T}g(k)$ and $\beta_{F}g(k)$ , the own-state indicator $s\in\{0,1\}$ (variant B only) with weight $\theta$ , and the field $c_{0}$ ; the choice probability is $\sigma(u)$ with $g(k)=1/(1+\gamma k)$ , $\gamma=0.9$ . (b) Observed correct-side rate in each of the 78 design cells of the 31,824 randomized queries (variant $\times$ $k$ $\times$ $l$ $\times$ $s$ ; 408 queries per cell; Wilson 95% intervals) against $u$ , the logit of the model’s mean predicted probability in the cell under divisive normalization with claim fixed effects. Color gives inbox size $k$ ; circles variant A, squares variant B. (c) The same cells under constant weights ( $g\equiv 1$ ): large- $k$ cells lie between the curve and 0.5 and small- $k$ cells beyond it. RMS deviation of cell rates from $\sigma(u)$ : 0.054 in (b), 0.118 in (c).

Figure 4: Outcome probability after eight rounds against the surrogate of the pre-registered rule (arm A), and low-

x_{0}

final fractions against surrogates of two coupling rules (arm B). (a) Observed outcome family

q(x_{0})

after eight rounds (solid, Wilson 95% intervals) against the eight-round surrogate of the blind pooled rule on the same graphs and seedings (dashed): the blind prediction is rejected; the correct-side outcome probability stays below 0.5 from every initial condition. (b) Discrimination arm: observed low-

x_{0}

final fractions against the eight-round surrogates of the normalized (divisive) and constant coupling rules with claim-resolved fields; the normalized rule lies closer, the constant rule is not rejected.

Figure 5: Final correct-side fractions for original, negated, and true claims. (a) Mean final correct-side fraction for the four original claims (denying side correct) and their minimally negated rewrites (affirming side correct) under identical conditions: the content-referenced outcome reverses with the wording. (b) True (SUPPORTS) claims converge on the correct = affirming side from all initial fractions.

Figure 6: qwen3:8b collective outcomes against predicted fixed points, and the per-message asymmetry at 70B. (a) qwen3:8b collectives: mean final correct-side fraction per cell; stars mark the predicted unstable fixed points, which coincide with the observed basin boundaries for claims 6, 1239, and 91569 at $\alpha=0.30$ ; claim 1569 is predicted and observed monostable correct-side. Cell values are mean final fractions; intermediate values are splits between the two basins, not intermediate end states (per-cell outcome counts in the SI). (b) The per-message assertion asymmetry on the same eight claims and message bank (claim-clustered 95% intervals) is not detected at 70B.

Figure 7: Calibration of transition-point predictions (qwen3:8b). Predicted unstable-fixed-point position (per-claim divisive response function) against the observed transition point (50% crossing of the eight-round outcome) for every (claim,

\alpha

) cell observed bistable: circles, the eight-claim campaign (11 cells); squares, the four-claim campaign (6 cells). Vertical bars are episode-bootstrap 95% intervals of the observed transition point, horizontal bars are single-agent-bootstrap 95% intervals of the prediction, and the dotted line is identity. Cells predicted and observed monostable (4 of 16 and 6 of 12) and the one class mismatch (claim 1146 at

\alpha=0.30

) are listed in the SI.

Term	Meaning
Agent	One LLM instance with a persona, holding a binary stance (TRUE or FALSE) on a claim; $N=32$ agents form a collective.
Claim; correct side	A short factual statement from CLIMATE-FEVER; the correct side is the side matching its label (mostly the denying side here, since most claims used are misinformation).
Inbox	The $k$ messages an agent reads in a round, $l$ of which argue for the correct side.
Message capacity $\alpha$ (crowding parameter)	An agent that has already accepted $r$ sources accepts the next with probability $e^{-\alpha r}$ ; larger $\alpha$ means fewer messages read (mean about $\ln(1+\alpha(N-1))/\alpha$ ; 6.4 sources at $\alpha=0.435$ ).
Response function	The probability that an agent answers on the correct side given its inbox; measured as a logistic function of a weighted sum of the messages.
Field $h$ (threshold)	The agent’s threshold for the correct side with its sign reversed: the log-odds of answering on the correct side with an empty inbox, set by the claim’s wording; positive means the wording alone lowers the threshold for the correct side. It shifts the logistic response; the steepness is set by the weights.

Table S1: Glossary of the terms used in the main text.

Figure S1: Identified per-message coefficients versus inbox size (31,824 randomized queries).

\beta_{T}(k)

and

\beta_{F}(k)

from the joint fit with claim fixed effects anchored by the empty-inbox cells, for prompt variants A (a) and B (b); bars are 95% claim-clustered bootstrap intervals.

claim	$\alpha$	predicted	pred. $x_{c}$ [95% CI] (frac. bistable)	observed	obs. crossing [95% CI]	$n_{\mathrm{c}}/n_{\mathrm{w}}/n_{\mathrm{u}}$
97	0.3	mono-correct	—	mono-correct	—	16/0/0
97	0.55	mono-correct	—	mono-correct	—	16/0/0
184	0.3	mono-correct	—	mono-correct	—	16/0/0
184	0.55	mono-correct	—	mono-correct	—	16/0/0
590	0.3	bistable	0.559 [0.497, 0.619] (1.00)	bistable	0.560 [0.465, 0.653]	7/9/0
590	0.55	bistable	0.578 [0.516, 0.637] (1.00)	bistable	0.622 [0.536, 0.701]	6/10/0

Table S2: All 16 cells of the eight-claim calibration campaign (qwen3:8b, 256 episodes, zero parse failures). Fifteen cells match in class; the mismatch is claim 1146 at

\alpha=0.30

, predicted bistable with threshold 0.094 just above the lowest seeded fraction (2/32), observed monostable correct. All eleven predicted thresholds of the cells observed bistable lie inside the observed intervals (mean absolute difference 0.04; Spearman

r_{s}=0.96

, claim-permutation

p=0.004

variant	tag (digest)	parsed	$\|b_{T}\|/\|b_{F}\|$	95% CI
instruct (control)	llama3.1:8b (46e0c10c039e)	5,984/5,984	1.437	[1.250, 1.651]
abliterated	mannix/llama3.1-8b-abliterated (f7d729db9832, Q4_0)	5,984/5,984	1.243	[1.051, 1.471]
base	llama3.1:8b-text-q4_K_M (6f98b5a6e4b7)	294/5,984	(2.10)	not evaluable

Table S3: Per-message assertion asymmetry (claim-fixed-effect logistic regression; claim-clustered bootstrap intervals). The instruct control reproduces the Stage-1 value 1.44 under the added format prefix. Bare affirmation indices

S

for the four original/negated pairs: instruct 0.92–1.52 (mean 1.22), abliterated 1.13–1.42 (mean 1.24); base cells parsed 2–6 of 64 and are not evaluable.

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